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<p>不是，该图展示的是空间复杂度，其反映的是增长趋势，而不是占用空间的绝对大小。</p>
<p>假设取 <span class="arithmatex">\(n = 8\)</span> ，你可能会发现每条曲线的值与函数对应不上。这是因为每条曲线都包含一个常数项，用于将取值范围压缩到一个视觉舒适的范围内。</p>
<p>在实际中，因为我们通常不知道每个方法的“常数项”复杂度是多少，所以一般无法仅凭复杂度来选择 <span class="arithmatex">\(n = 8\)</span> 之下的最优解法。但对于 <span class="arithmatex">\(n = 8^5\)</span> 就很好选了，这时增长趋势已经占主导了。</p>
<p><strong>Q</strong> 是否存在根据实际使用场景，选择牺牲时间（或空间）来设计算法的情况？</p>
<p>在实际应用中，大部分情况会选择牺牲空间换时间。例如数据库索引，我们通常选择建立 B+ 树或哈希索引，占用大量内存空间，以换取 <span class="arithmatex">\(O(\log n)\)</span> 甚至 <span class="arithmatex">\(O(1)\)</span> 的高效查询。</p>
<p>在空间资源宝贵的场景，也会选择牺牲时间换空间。例如在嵌入式开发中，设备内存很宝贵，工程师可能会放弃使用哈希表，选择使用数组顺序查找，以节省内存占用，代价是查找变慢。</p>
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<p>输入一个 <code>key</code> ，哈希函数的计算过程分为以下两步。</p>
<ol>
<li>通过某种哈希算法 <code>hash()</code> 计算得到哈希值。</li>
<li>将哈希值对桶数量（数组长度）<code>capacity</code> 取模，从而获取该 <code>key</code> 对应的数组索引 <code>index</code> 。</li>
<li>将哈希值对桶数量（数组长度）<code>capacity</code> 取模，从而获取该 <code>key</code> 对应的桶（数组索引）<code>index</code> 。</li>
</ol>
<div class="highlight"><pre><span></span><code><a id="__codelineno-28-1" name="__codelineno-28-1" href="#__codelineno-28-1"></a><span class="nv">index</span><span class="w"> </span><span class="o">=</span><span class="w"> </span>hash<span class="o">(</span>key<span class="o">)</span><span class="w"> </span>%<span class="w"> </span>capacity
</code></pre></div>
@@ -4229,7 +4229,7 @@
<a id="__codelineno-29-18" name="__codelineno-29-18" href="#__codelineno-29-18"></a> <span class="n">index</span> <span class="o">=</span> <span class="n">key</span> <span class="o">%</span> <span class="mi">100</span>
<a id="__codelineno-29-19" name="__codelineno-29-19" href="#__codelineno-29-19"></a> <span class="k">return</span> <span class="n">index</span>
<a id="__codelineno-29-20" name="__codelineno-29-20" href="#__codelineno-29-20"></a>
<a id="__codelineno-29-21" name="__codelineno-29-21" href="#__codelineno-29-21"></a> <span class="k">def</span><span class="w"> </span><span class="nf">get</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">key</span><span class="p">:</span> <span class="nb">int</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">str</span><span class="p">:</span>
<a id="__codelineno-29-21" name="__codelineno-29-21" href="#__codelineno-29-21"></a> <span class="k">def</span><span class="w"> </span><span class="nf">get</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">key</span><span class="p">:</span> <span class="nb">int</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">str</span> <span class="o">|</span> <span class="kc">None</span><span class="p">:</span>
<a id="__codelineno-29-22" name="__codelineno-29-22" href="#__codelineno-29-22"></a><span class="w"> </span><span class="sd">&quot;&quot;&quot;查询操作&quot;&quot;&quot;</span>
<a id="__codelineno-29-23" name="__codelineno-29-23" href="#__codelineno-29-23"></a> <span class="n">index</span><span class="p">:</span> <span class="nb">int</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">hash_func</span><span class="p">(</span><span class="n">key</span><span class="p">)</span>
<a id="__codelineno-29-24" name="__codelineno-29-24" href="#__codelineno-29-24"></a> <span class="n">pair</span><span class="p">:</span> <span class="n">Pair</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">buckets</span><span class="p">[</span><span class="n">index</span><span class="p">]</span>
@@ -4238,7 +4238,7 @@
<a id="__codelineno-29-27" name="__codelineno-29-27" href="#__codelineno-29-27"></a> <span class="k">return</span> <span class="n">pair</span><span class="o">.</span><span class="n">val</span>
<a id="__codelineno-29-28" name="__codelineno-29-28" href="#__codelineno-29-28"></a>
<a id="__codelineno-29-29" name="__codelineno-29-29" href="#__codelineno-29-29"></a> <span class="k">def</span><span class="w"> </span><span class="nf">put</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">key</span><span class="p">:</span> <span class="nb">int</span><span class="p">,</span> <span class="n">val</span><span class="p">:</span> <span class="nb">str</span><span class="p">):</span>
<a id="__codelineno-29-30" name="__codelineno-29-30" href="#__codelineno-29-30"></a><span class="w"> </span><span class="sd">&quot;&quot;&quot;添加操作&quot;&quot;&quot;</span>
<a id="__codelineno-29-30" name="__codelineno-29-30" href="#__codelineno-29-30"></a><span class="w"> </span><span class="sd">&quot;&quot;&quot;添加和更新操作&quot;&quot;&quot;</span>
<a id="__codelineno-29-31" name="__codelineno-29-31" href="#__codelineno-29-31"></a> <span class="n">pair</span> <span class="o">=</span> <span class="n">Pair</span><span class="p">(</span><span class="n">key</span><span class="p">,</span> <span class="n">val</span><span class="p">)</span>
<a id="__codelineno-29-32" name="__codelineno-29-32" href="#__codelineno-29-32"></a> <span class="n">index</span><span class="p">:</span> <span class="nb">int</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">hash_func</span><span class="p">(</span><span class="n">key</span><span class="p">)</span>
<a id="__codelineno-29-33" name="__codelineno-29-33" href="#__codelineno-29-33"></a> <span class="bp">self</span><span class="o">.</span><span class="n">buckets</span><span class="p">[</span><span class="n">index</span><span class="p">]</span> <span class="o">=</span> <span class="n">pair</span>
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<p>查找的时候通过哈希函数找到对应的桶和键值对，发现 <code>key</code> 不匹配，这就代表有哈希冲突。因此，线性探测法会根据预先设定的步长依次向下查找，直至找到正确的键值对或无法找到跳出为止。</p>
<p><strong>Q</strong>：为什么哈希表扩容能够缓解哈希冲突？</p>
<p>哈希函数的最后一步往往是对数组长度 <span class="arithmatex">\(n\)</span> 取模（取余），让输出值落在数组索引范围内；在扩容后，数组长度 <span class="arithmatex">\(n\)</span> 发生变化，而 <code>key</code> 对应的索引也可能发生变化。原先落在同一个桶的多个 <code>key</code> ，在扩容后可能会被分配到多个桶中，从而实现哈希冲突的缓解。</p>
<p><strong>Q</strong>：如果为了高效的存取，那么直接使用数组不就好了吗？</p>
<p>当数据的 <code>key</code> 是连续的小范围整数时，直接用数组即可，简单高效。但当 <code>key</code> 是其他类型（例如字符串）时，就需要借助哈希函数将 <code>key</code> 映射为数组索引，再通过桶数组存储元素，这样的结构就是哈希表。</p>
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<h1 id="51">5.1 &nbsp; 栈<a class="headerlink" href="#51" title="Permanent link">&para;</a></h1>
<p><u>栈（stack）</u>是一种遵循先入后出逻辑的线性数据结构。</p>
<p>我们可以将栈类比为桌面上的一摞盘子，如果想取出底部的盘子，则需要先将上面的盘子依次移走。我们将盘子替换为各种类型的元素（如整数、字符、对象等），就得到了栈这种数据结构。</p>
<p>我们可以将栈类比为桌面上的一摞盘子，规定每次只能移动一个盘子，那么想取出底部的盘子，则需要先将上面的盘子依次移走。我们将盘子替换为各种类型的元素（如整数、字符、对象等），就得到了栈这种数据结构。</p>
<p>如图 5-1 所示，我们把堆叠元素的顶部称为“栈顶”，底部称为“栈底”。将把元素添加到栈顶的操作叫作“入栈”，删除栈顶元素的操作叫作“出栈”。</p>
<p><a class="glightbox" href="../stack.assets/stack_operations.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="栈的先入后出规则" class="animation-figure" src="../stack.assets/stack_operations.png" /></a></p>
<p align="center"> 图 5-1 &nbsp; 栈的先入后出规则 </p>
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<p><strong>The logical structures reveal the logical relationships between data elements</strong>. In arrays and linked lists, data are arranged in a specific sequence, demonstrating the linear relationship between data; while in trees, data are arranged hierarchically from the top down, showing the derived relationship between "ancestors" and "descendants"; and graphs are composed of nodes and edges, reflecting the intricate network relationship.</p>
<p>As shown in Figure 3-1, logical structures can be divided into two major categories: "linear" and "non-linear". Linear structures are more intuitive, indicating data is arranged linearly in logical relationships; non-linear structures, conversely, are arranged non-linearly.</p>
<ul>
<li><strong>Linear data structures</strong>: Arrays, Linked Lists, Stacks, Queues, Hash Tables.</li>
<li><strong>Linear data structures</strong>: Arrays, Linked Lists, Stacks, Queues, Hash Tables, where elements have a one-to-one sequential relationship.</li>
<li><strong>Non-linear data structures</strong>: Trees, Heaps, Graphs, Hash Tables.</li>
</ul>
<p>Non-linear data structures can be further divided into tree structures and network structures.</p>
@@ -3608,7 +3608,7 @@
<p class="admonition-title">Tip</p>
<p>It's worth noting that comparing memory to an Excel spreadsheet is a simplified analogy. The actual working mechanism of memory is more complex, involving concepts like address space, memory management, cache mechanisms, virtual memory, and physical memory.</p>
</div>
<p>Memory is a shared resource for all programs. When a block of memory is occupied by one program, it cannot be simultaneously used by other programs. <strong>Therefore, considering memory resources is crucial in designing data structures and algorithms</strong>. For instance, the algorithm's peak memory usage should not exceed the remaining free memory of the system; if there is a lack of contiguous memory blocks, then the data structure chosen must be able to be stored in non-contiguous memory blocks.</p>
<p>Memory is a shared resource for all programs. When a block of memory is occupied by one program, it cannot be simultaneously used by other programs. <strong>Therefore, memory resources are an important consideration in the design of data structures and algorithms</strong>. For instance, the algorithm's peak memory usage should not exceed the remaining free memory of the system; if there is a lack of contiguous memory blocks, then the data structure chosen must be able to be stored in non-contiguous memory blocks.</p>
<p>As illustrated in Figure 3-3, <strong>the physical structure reflects the way data is stored in computer memory</strong> and it can be divided into contiguous space storage (arrays) and non-contiguous space storage (linked lists). The two types of physical structures exhibit complementary characteristics in terms of time efficiency and space efficiency.</p>
<p><a class="glightbox" href="../classification_of_data_structure.assets/classification_phisical_structure.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="Contiguous space storage and dispersed space storage" class="animation-figure" src="../classification_of_data_structure.assets/classification_phisical_structure.png" /></a></p>
<p align="center"> Figure 3-3 &nbsp; Contiguous space storage and dispersed space storage </p>
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<p>不是，該圖展示的是空間複雜度，其反映的是增長趨勢，而不是佔用空間的絕對大小。</p>
<p>假設取 <span class="arithmatex">\(n = 8\)</span> ，你可能會發現每條曲線的值與函式對應不上。這是因為每條曲線都包含一個常數項，用於將取值範圍壓縮到一個視覺舒適的範圍內。</p>
<p>在實際中，因為我們通常不知道每個方法的“常數項”複雜度是多少，所以一般無法僅憑複雜度來選擇 <span class="arithmatex">\(n = 8\)</span> 之下的最優解法。但對於 <span class="arithmatex">\(n = 8^5\)</span> 就很好選了，這時增長趨勢已經佔主導了。</p>
<p><strong>Q</strong> 是否存在根據實際使用場景，選擇犧牲時間（或空間）來設計演算法的情況？</p>
<p>在實際應用中，大部分情況會選擇犧牲空間換時間。例如資料庫索引，我們通常選擇建立 B+ 樹或雜湊索引，佔用大量記憶體空間，以換取 <span class="arithmatex">\(O(\log n)\)</span> 甚至 <span class="arithmatex">\(O(1)\)</span> 的高效查詢。</p>
<p>在空間資源寶貴的場景，也會選擇犧牲時間換空間。例如在嵌入式開發中，裝置記憶體很寶貴，工程師可能會放棄使用雜湊表，選擇使用陣列順序查詢，以節省記憶體佔用，代價是查詢變慢。</p>
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<p>輸入一個 <code>key</code> ，雜湊函式的計算過程分為以下兩步。</p>
<ol>
<li>透過某種雜湊演算法 <code>hash()</code> 計算得到雜湊值。</li>
<li>將雜湊值對桶數量（陣列長度）<code>capacity</code> 取模，從而獲取該 <code>key</code> 對應的陣列索引 <code>index</code> 。</li>
<li>將雜湊值對桶數量（陣列長度）<code>capacity</code> 取模，從而獲取該 <code>key</code> 對應的桶（陣列索引）<code>index</code> 。</li>
</ol>
<div class="highlight"><pre><span></span><code><a id="__codelineno-28-1" name="__codelineno-28-1" href="#__codelineno-28-1"></a><span class="nv">index</span><span class="w"> </span><span class="o">=</span><span class="w"> </span>hash<span class="o">(</span>key<span class="o">)</span><span class="w"> </span>%<span class="w"> </span>capacity
</code></pre></div>
@@ -4172,7 +4172,7 @@
<a id="__codelineno-29-18" name="__codelineno-29-18" href="#__codelineno-29-18"></a> <span class="n">index</span> <span class="o">=</span> <span class="n">key</span> <span class="o">%</span> <span class="mi">100</span>
<a id="__codelineno-29-19" name="__codelineno-29-19" href="#__codelineno-29-19"></a> <span class="k">return</span> <span class="n">index</span>
<a id="__codelineno-29-20" name="__codelineno-29-20" href="#__codelineno-29-20"></a>
<a id="__codelineno-29-21" name="__codelineno-29-21" href="#__codelineno-29-21"></a> <span class="k">def</span><span class="w"> </span><span class="nf">get</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">key</span><span class="p">:</span> <span class="nb">int</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">str</span><span class="p">:</span>
<a id="__codelineno-29-21" name="__codelineno-29-21" href="#__codelineno-29-21"></a> <span class="k">def</span><span class="w"> </span><span class="nf">get</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">key</span><span class="p">:</span> <span class="nb">int</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">str</span> <span class="o">|</span> <span class="kc">None</span><span class="p">:</span>
<a id="__codelineno-29-22" name="__codelineno-29-22" href="#__codelineno-29-22"></a><span class="w"> </span><span class="sd">&quot;&quot;&quot;查詢操作&quot;&quot;&quot;</span>
<a id="__codelineno-29-23" name="__codelineno-29-23" href="#__codelineno-29-23"></a> <span class="n">index</span><span class="p">:</span> <span class="nb">int</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">hash_func</span><span class="p">(</span><span class="n">key</span><span class="p">)</span>
<a id="__codelineno-29-24" name="__codelineno-29-24" href="#__codelineno-29-24"></a> <span class="n">pair</span><span class="p">:</span> <span class="n">Pair</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">buckets</span><span class="p">[</span><span class="n">index</span><span class="p">]</span>
@@ -4181,7 +4181,7 @@
<a id="__codelineno-29-27" name="__codelineno-29-27" href="#__codelineno-29-27"></a> <span class="k">return</span> <span class="n">pair</span><span class="o">.</span><span class="n">val</span>
<a id="__codelineno-29-28" name="__codelineno-29-28" href="#__codelineno-29-28"></a>
<a id="__codelineno-29-29" name="__codelineno-29-29" href="#__codelineno-29-29"></a> <span class="k">def</span><span class="w"> </span><span class="nf">put</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">key</span><span class="p">:</span> <span class="nb">int</span><span class="p">,</span> <span class="n">val</span><span class="p">:</span> <span class="nb">str</span><span class="p">):</span>
<a id="__codelineno-29-30" name="__codelineno-29-30" href="#__codelineno-29-30"></a><span class="w"> </span><span class="sd">&quot;&quot;&quot;新增操作&quot;&quot;&quot;</span>
<a id="__codelineno-29-30" name="__codelineno-29-30" href="#__codelineno-29-30"></a><span class="w"> </span><span class="sd">&quot;&quot;&quot;新增和更新操作&quot;&quot;&quot;</span>
<a id="__codelineno-29-31" name="__codelineno-29-31" href="#__codelineno-29-31"></a> <span class="n">pair</span> <span class="o">=</span> <span class="n">Pair</span><span class="p">(</span><span class="n">key</span><span class="p">,</span> <span class="n">val</span><span class="p">)</span>
<a id="__codelineno-29-32" name="__codelineno-29-32" href="#__codelineno-29-32"></a> <span class="n">index</span><span class="p">:</span> <span class="nb">int</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">hash_func</span><span class="p">(</span><span class="n">key</span><span class="p">)</span>
<a id="__codelineno-29-33" name="__codelineno-29-33" href="#__codelineno-29-33"></a> <span class="bp">self</span><span class="o">.</span><span class="n">buckets</span><span class="p">[</span><span class="n">index</span><span class="p">]</span> <span class="o">=</span> <span class="n">pair</span>
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<p>查詢的時候透過雜湊函式找到對應的桶和鍵值對，發現 <code>key</code> 不匹配，這就代表有雜湊衝突。因此，線性探查法會根據預先設定的步長依次向下查詢，直至找到正確的鍵值對或無法找到跳出為止。</p>
<p><strong>Q</strong>：為什麼雜湊表擴容能夠緩解雜湊衝突？</p>
<p>雜湊函式的最後一步往往是對陣列長度 <span class="arithmatex">\(n\)</span> 取模（取餘），讓輸出值落在陣列索引範圍內；在擴容後，陣列長度 <span class="arithmatex">\(n\)</span> 發生變化，而 <code>key</code> 對應的索引也可能發生變化。原先落在同一個桶的多個 <code>key</code> ，在擴容後可能會被分配到多個桶中，從而實現雜湊衝突的緩解。</p>
<p><strong>Q</strong>：如果為了高效的存取，那麼直接使用陣列不就好了嗎？</p>
<p>當資料的 <code>key</code> 是連續的小範圍整數時，直接用陣列即可，簡單高效。但當 <code>key</code> 是其他型別（例如字串）時，就需要藉助雜湊函式將 <code>key</code> 對映為陣列索引，再透過桶陣列儲存元素，這樣的結構就是雜湊表。</p>
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<!-- Page content -->
<h1 id="51">5.1 &nbsp; 堆疊<a class="headerlink" href="#51" title="Permanent link">&para;</a></h1>
<p><u>堆疊（stack）</u>是一種遵循先入後出邏輯的線性資料結構。</p>
<p>我們可以將堆疊類比為桌面上的一疊盤子，如果想取出底部的盤子，則需要先將上面的盤子依次移走。我們將盤子替換為各種型別的元素（如整數、字元、物件等），就得到了堆疊這種資料結構。</p>
<p>我們可以將堆疊類比為桌面上的一疊盤子，規定每次只能移動一個盤子，那麼想取出底部的盤子，則需要先將上面的盤子依次移走。我們將盤子替換為各種型別的元素（如整數、字元、物件等），就得到了堆疊這種資料結構。</p>
<p>如圖 5-1 所示，我們把堆積疊元素的頂部稱為“堆疊頂”，底部稱為“堆疊底”。將把元素新增到堆疊頂的操作叫作“入堆疊”，刪除堆疊頂元素的操作叫作“出堆疊”。</p>
<p><a class="glightbox" href="../stack.assets/stack_operations.png" data-type="image" data-width="100%" data-height="auto" data-desc-position="bottom"><img alt="堆疊的先入後出規則" class="animation-figure" src="../stack.assets/stack_operations.png" /></a></p>
<p align="center"> 圖 5-1 &nbsp; 堆疊的先入後出規則 </p>
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